Degrees containing members of thin Π10 classes are dense and co-dense

In [Countable thin Π10 classes, Ann. Pure Appl. Logic 59 (1993) 79–139], Cenzer, Downey, Jockusch and Shore proved the density of degrees (not necessarily c.e.) containing members of countable thin Π10 classes. In the same paper, Cenzer et al. also proved the existence of degrees containing no members of thin Π10 classes. We will prove in this paper that the c.e. degrees containing no members of thin Π10 classes are dense in the c.e. degrees. We will also prove that the c.e. degrees containing members of thin Π10 classes are dense in the c.e. degrees, improving the result of Cenzer et al. mentioned above. Thus, we obtain a new natural subclass of c.e. degrees which are both dense and co-dense in the c.e. degrees, while the other such class is the class of branching c.e. degrees (See [P. Fejer, The density of the nonbranching degrees, Ann. Pure Appl. Logic 24 (1983) 113–130] for nonbranching degrees and [T. A. Slaman, The density of infima in the recursively enumerable degrees, Ann. Pure Appl. Logic 52 (19...